In four dimensions the difference between a metric and an Einstein metric
is measured by the trace-free Ricci tensor E_g = Ric_g - (Scal_g/4) g. Its
L^2 norm squared, D(g) = int_M |E_g|_g^2 dmu_g, vanishes exactly for
Einstein metrics and is invariant under constant rescaling of g. The
question is whether the exact-Einstein classification of certain
four-manifolds persists to metrics with small but nonzero D, under
nonnegative sectional curvature.
Main theorem. There is a universal constant epsilon_0 > 0 with the
following property. Let M be a closed connected simply connected
four-manifold with a smooth Riemannian metric g satisfying sec_g >= 0 and
D(g) < epsilon_0. Then M is diffeomorphic to one of S^4, CP^2, or
S^2 x S^2.
The conclusion is about the underlying smooth manifold, not the metric.
No auxiliary bounds on curvature, volume, diameter, injectivity radius,
Sobolev constants, or curvature derivatives are assumed. Small L^2 error
does not give pointwise curvature control, so this is not immediate from
the Einstein case.
Assumption (zero-plane rigidity). Let (N,h) be a closed connected
four-manifold with Ric_h = 3h and sec_h >= 0. Then the universal
Riemannian cover of (N,h) is isometric to one of: the round S^4 of
sectional curvature 1; the Fubini-Study CP^2 normalized so Ric = 3g; or
S^2(1/sqrt3) x S^2(1/sqrt3) with the product metric.
This assumption is the exact-Einstein classification for the metric with
nonnegative sectional curvature. Partial results in the direction include
Gursky-LeBrun for metrics with nonzero positive-definite intersection
form, Cao-Tran under sectional pinching, and Liu for the T^2-invariant
case. The classification itself is not proved here; only the consequence
for small D is.
Gursky-LeBrun, On Einstein manifolds of positive sectional curvature,
Ann. Global Anal. Geom. 17 (1999) 315-328.
https://arxiv.org/pdf/math/9807055
Cao-Tran, Einstein four-manifolds of pinched sectional curvature,
Adv. Math. 335 (2018) 322-342.
https://doi.org/10.1016/j.aim.2018.07.003
Liu, Toric Einstein 4-manifolds with non-negative sectional curvature,
Ann. Global Anal. Geom. 67 (2025) 14.
https://doi.org/10.1007/s10455-025-09990-3
Three observations frame the proof. First, D is a global L^2 quantity,
so it gives no pointwise bound. A curvature spike on a region of tiny
volume can contribute arbitrarily little to D while being locally huge.
Second, Ricci flow can in principle develop a singularity before any
local defect has spread. Third, sec_g >= 0 is a hypothesis at time zero
only; the flow may produce negative sectional curvature. The three
problems are addressed by a single high-curvature tail bound.
Strategy. Rescale so the average scalar curvature equals 12, which is
free because D is scale-invariant. Derive uniform volume and diameter
bounds using the Euler characteristic, an algebraic comparison between
sectional and Ricci curvatures, the almost-Schur inequality, and a
Myers-style second variation argument along many geodesics. Derive a
uniform Sobolev inequality via a dyadic truncation trick. Run Ricci flow
for a fixed universal time T; entropy monotonicity and the high-curvature
tail bound give the Sobolev inequality, noncollapse, and uniform lifespan
along the flow. Propagate the Einstein defect in L^2, and bound the
integral of the negative part of sectional curvature by Ct. Extract a
compact limit by Hamilton's compactness theorem. The limit is an exact
shrinking Einstein metric with nonnegative sectional curvature; the
Einstein classification identifies it, and the classification transfers
back to M by diffeomorphism.
Throughout, int_M F dmu denotes the Riemannian integral over M and
||T||_p denotes the L^p(g) norm of a tensor T, pointwise computed in an
orthonormal frame. Constants C and epsilon are dimension-four constants,
freely changed from line to line. Symbols chi and b_i are topological
invariants of M.
Curvature signs are fixed so the unit round S^4 has Rm(X,Y,X,Y) = 1 for
orthonormal X, Y. The Ricci tensor is the trace of Rm:
Ric(X,X) = sum_i Rm(e_i, X, X, e_i) over any orthonormal frame. The
scalar curvature is R = sum_i Ric(e_i, e_i). When sec_g >= 0, the
section-sum identity
R = 2 sum_{i < j} sec(e_i, e_j) (1)
implies R >= 0 everywhere, and each sectional curvature is bounded
above by R/2. On the finite-dimensional space of algebraic curvature
tensors over R^4, all norms are equivalent. Taking |B| as the
sum-of-squares norm in an orthonormal frame yields a dimension-four
constant C_0 with
|Rm| <= C_0 R (2)
at every point where sec >= 0.
The Laplacian is Delta = tr(nabla^2). On a closed manifold Delta is
negative semidefinite, int_M (Delta f) f dmu = -int_M |nabla f|^2 dmu,
and the equation Delta f = u with int_M u dmu = 0 has a unique smooth
solution with int_M f dmu = 0.
Initial bounds. After constant rescaling, there are uniform positive
constants v_0, V_0, D_0 such that
(1/V) int_M R dmu = 12, (3a)
||R - 12||_{L^2} <= 4 ||E||_{L^2}, (3b)
v_0 <= V <= V_0, (3c)
diam(M, g) <= D_0, (3d)
||Ric - 3g||_{L^2} -> 0. (3e)
Proof. Simple connectivity and Poincare duality give b_0 = b_4 = 1 and
b_1 = b_3 = 0, so the Euler characteristic is
chi(M) = 2 + b_2(M) >= 2. (4)
The Chern-Gauss-Bonnet formula reads
chi(M) = (1/(8 pi^2)) int_M (|W|^2 - |rho|^2/2 + R^2/24) dmu,
where W is the Weyl tensor and rho = E is trace-free. Orthogonally,
|Rm|^2 = |W|^2 + 2|rho|^2 + R^2/6, so each term of the Gauss-Bonnet
integrand is dominated by |Rm|^2 with universal constant. Thus
chi(M) <= C int_M |Rm|^2 dmu. (5)
If (1/V) int_M R dmu = 0, then R = 0 everywhere by (1), so Rm = 0 by
(2), contradicting (4) and (5). The average scalar curvature is thus
positive. Rescaling by a constant achieves (3a); the defect D is
unchanged because it is scale-invariant in dimension four.
The almost-Schur inequality of De Lellis and Topping gives (3b). Solve
Delta f = R - 12 with int_M f dmu = 0. The contracted Bianchi identity
nabla^i Ric_ij = (1/2) d_j R, combined with Ric = E + (R/4) g, implies
div E = (1/4) dR. Two integrations by parts give
int_M (R-12)^2 dmu = int_M (R-12) Delta f dmu
= -int_M g(nabla R, nabla f) dmu
= -4 int_M g(div E, nabla f) dmu
= 4 int_M <E, Hess f> dmu. (6)
Cauchy-Schwarz bounds the right by 4 ||E||_{L^2} ||Hess f||_{L^2}. The
Bochner formula, together with Ric >= 0 (which follows from sec >= 0),
gives
int_M |Hess f|^2 = int_M (Delta f)^2 - int_M Ric(nabla f, nabla f)
<= int_M (R - 12)^2. (7)
Substituting (7) into (6) and dividing by ||R - 12||_{L^2} yields (3b).
The splitting Ric - 3g = E + (R - 12) g/4 is orthogonal in the trace
decomposition, so
||Ric - 3g||_{L^2}^2 = ||E||^2 + (1/4) ||R - 12||^2
<= 5 ||E||^2 -> 0, (8)
which is (3e).
De Lellis and Topping, The almost-Schur lemma, Calc. Var. PDE 43 (2012)
347-354.
https://doi.org/10.1007/s00526-011-0413-z
Lower volume. From (3a), int_M R^2 dmu = int_M (R-12)^2 dmu + 144 V.
Combined with (5) and (2):
2 <= chi(M) <= C int_M |Rm|^2 dmu <= C C_0^2 int_M R^2 dmu
<= C C_0^2 (16 ||E||^2 + 144 V).
For large j the ||E||^2 term is negligible, so V >= v_0 > 0 with v_0
universal. This is the lower half of (3c).
Diameter. Let D = diam(M,g) and choose p, q realizing it. Let
A_1 = B(p, D/4) and A_2 = B(q, D/4). Bishop-Gromov on a closed
manifold with Ric >= 0 gives Vol(A_i) >= 8^{-4} V. For any continuous
H >= 0, the segment inequality of Cheeger and Colding (Theorem 2.11)
gives
int_{A_1 x A_2} int_{gamma_xy} H ds dmu(x) dmu(y)
<= C D (Vol(A_1) + Vol(A_2)) int_M H dmu, (9)
where gamma_xy is the minimizing geodesic from x to y, defined for
almost every pair.
Cheeger and Colding, Lower bounds on Ricci curvature and the almost
rigidity of warped products, Annals of Mathematics 144 (1996) 189-237.
https://doi.org/10.2307/2118589
Take H = |Ric - 3g|. Since int_M H dmu <= V^{1/2} ||Ric - 3g||_{L^2}
-> 0 by (3e), and Vol(A_1 x A_2) >= 8^{-8} V^2, some minimizing
geodesic gamma between A_1 and A_2 satisfies
ell := length(gamma) >= D/2, int_gamma |Ric - 3g| ds <= delta D, (10)
with delta -> 0.
Three parallel orthonormal normal fields n_1, n_2, n_3 along gamma,
multiplied by sin(pi s / ell), enter the index form. Second variation
of a minimizing geodesic gives
sum_{i=1}^3 int_0^ell (|V_i'|^2 - Rm(n_i, gamma', gamma', n_i)) ds >= 0. (11)
Each |V_i'|^2 = (pi/ell)^2 cos^2(pi s / ell) integrates to
pi^2 / (2 ell), and sum_i Rm(n_i, gamma', gamma', n_i) equals
Ric(gamma', gamma'). Write Ric = 3 g + (Ric - 3g), apply (10), and
integrate to get
3 pi^2 / (2 ell) >= (3/2) ell - delta D. (12)
Since ell >= D/2, this implies (3/2 - 2 delta) ell^2 <= 3 pi^2/2. For
small delta, ell <= pi sqrt(2) and D = 2 ell <= 2 pi sqrt(2). Setting
D_0 = 3 pi covers all sufficiently large j.
Upper volume. Bishop-Gromov with Ric >= 0 gives Vol B(x, r) <= omega_4
r^4 for all x and r, where omega_4 = pi^2/2 is the Euclidean unit
4-ball volume. With r = D and D <= D_0, V <= omega_4 D_0^4, giving the
upper half of (3c) with V_0 = omega_4 D_0^4.
Sobolev inequality. A dyadic truncation argument reduces a uniform
Sobolev inequality to control of the L^2-embeddings of functions of
small support.
Lemma 1 (Truncation). Let (M,g) be a closed four-manifold, a > 0 and
P >= 0 smooth, with E(f) = int_M (a |nabla f|^2 + P f^2) dmu. Suppose
that for every Lipschitz f with m = Vol{supp f} one has
||f||_{L^2}^2 <= C m^{1/2} E(f). (13)
Then there is C' depending only on C with
||f||_{L^4}^2 <= C' E(f). (14)
Proof. For k in Z, set m_k = Vol{|f| > 2^k} and f_k = min{(|f| - 2^k)_+,
2^k}. The hypothesis gives
2^{2k} m_{k+1} <= ||f_k||_{L^2}^2 <= C m_k^{1/2} E(f_k). (15)
Now |nabla f_k| = |nabla |f|| on the annulus 2^k < |f| < 2^{k+1} and
zero elsewhere, so the gradients of distinct f_k live on disjoint
annuli and sum_k ||nabla f_k||_{L^2}^2 = ||nabla f||_{L^2}^2. Also
f_k <= 2^k and sum_{2^k < |f|} 2^{2k} <= (4/3) |f|^2 pointwise, so
sum_k f_k^2 <= (4/3) f^2. Combined, sum_k E(f_k) <= C_2 E(f) for a
numerical C_2. Put b_k = 2^{4k} m_k. Then (15) reads
b_{k+1} <= 16 C b_k^{1/2} E(f_k). (16)
The sum B = sum_k b_k is comparable to ||f||_{L^4}^4. Indeed the
pointwise bound sum_{2^k < |f(x)|} 2^{4k} <= (16/15) |f(x)|^4
integrates to B <= (16/15) ||f||_{L^4}^4, and the reverse inequality
B >= (1/16) ||f||_{L^4}^4 holds by dropping all terms but the leading
one. Summing (16) over k and using b_k^{1/2} <= B^{1/2},
B <= 16 C_2 C B^{1/2} E(f). (17)
If B = 0 then f = 0. Otherwise B^{1/2} <= 16 C_2 C E(f), and
B <= (16 C_2 C)^2 E(f)^2. Combined with the lower bound on B,
||f||_{L^4}^2 <= C' E(f), (18)
with C' = 16 C_2 C sqrt(16/15).
Initial Sobolev. On the normalized metrics of (3), the inequality
||f||_{L^4}^2 <= S int_M (|nabla f|^2 + f^2) dmu (19)
holds for every Lipschitz f, with a uniform S.
Proof. Apply Lemma 1 with a = P = 1. It suffices to establish (13) for
small m.
For Lipschitz f with mean f_bar = (1/V) int_M f dmu, define the kernel
K(x, y) = d(x, y)^{-3} on M x M minus the diagonal. Integrating
|f(x) - f(y)| along the minimizing geodesic from x to y and averaging
in polar coordinates gives, using the Jacobian comparison
J_x(r, theta) <= (r/s)^3 J_x(s, theta) <= 8 J_x(s, theta) (20)
on the half-segment r/2 <= s <= r and the reverse on r/2 <= s:
|f(x) - f_bar| <= (D^4 / (4V)) int_M |nabla f(y)| d(x, y)^{-3} dmu(y). (21)
Bishop comparison for Ric >= 0 (Vol B(x, r) <= omega_4 r^4, and
Vol of a sphere of radius r <= 4 omega_4 r^3) gives the kernel bound:
for any measurable A of volume m > 0,
sup_{x in A} int_A d(x, y)^{-3} dmu(y) <= C m^{1/4}. (22)
Proof of (22). Fix x. Set rho = m^{1/4}. On A cap B(x, rho), the
integral is at most int_0^rho r^{-3} 4 omega_4 r^3 dr = 4 omega_4 rho
since Vol of the geodesic sphere of radius r is at most 4 omega_4 r^3.
On A cap {d(x, y) > rho}, the integrand is at most rho^{-3} and the
volume is m. Adding the two bounds gives (4 omega_4 + 1) m^{1/4}.
Schur test. The integral operator T on L^2(A) with kernel
K(x, y) = d(x, y)^{-3} has operator norm at most C m^{1/4}, by
applying the Schur test to (22) in both variables.
Endgame. Take A = supp f. Applying (21) and the Schur operator bound,
||f - f_bar||_{L^2(A)} <= (C D^4 / V) m^{1/4} ||nabla f||_{L^2}. (23)
On A write f = (f - f_bar) + f_bar, and note sqrt(m) |f_bar| <=
(m/V) ||f||_{L^2}. Then
||f||_{L^2} <= (C D^4 / V) m^{1/4} ||nabla f||_{L^2}
+ (m/V) ||f||_{L^2}. (24)
If m/V <= 1/2, absorb the second term and use D <= D_0 and V >= v_0:
||f||_{L^2}^2 <= C m^{1/2} ||nabla f||_{L^2}^2 <= C m^{1/2} E(f). (25)
If m/V > 1/2, then m >= v_0/2 and E(f) >= ||f||_{L^2}^2, so
||f||_{L^2}^2 <= E(f) <= (2/v_0)^{1/2} m^{1/2} E(f). (26)
In either case (13) holds, and Lemma 1 gives (19).
Ricci flow. The parabolic equation
partial_t g(t) = -2 Ric(g(t)), g(0) = g, (27)
has a unique smooth solution on a maximal interval [0, T_max), by
Hamilton's short-time existence theorem. The scalar curvature evolves
by partial_t R = Delta R + 2|Ric|^2, so R >= 0 is preserved by the
maximum principle when R(0) >= 0. The volume obeys partial_t dmu =
-R dmu, hence decreases. Shi's interior derivative estimate applies:
if |Rm| <= K on a parabolic slab of unit dimension, then
|nabla Rm| <= C K^{3/2} in the interior, with a derivative-independent
constant.
Hamilton, Three-manifolds with positive Ricci curvature,
J. Differential Geom. 17 (1982) 255-306.
https://doi.org/10.4310/jdg/1214436922
Shi, Deforming the metric on complete Riemannian manifolds,
J. Differential Geom. 30 (1989) 223-301.
https://doi.org/10.4310/jdg/1214443292
Perelman's entropy. In dimension four, define
mu(g, tau) = inf_{||f||_{L^2} = 1} [ tau int_M (4|nabla f|^2 + R f^2) dmu
- int_M f^2 log f^2 dmu - 2 log(4 pi tau) - 4 ], (28)
the infimum over smooth positive f of unit L^2 norm. Perelman's
monotonicity gives, for any tau > 0 and any 0 <= t < T_max,
mu(g(t), tau) >= mu(g(0), tau + t). (29)
The entropy parameter on the right is tau + t, not a flow time; only
g(0) enters explicitly. So (29) does not restrict T_max.
Perelman, The entropy formula for the Ricci flow and its geometric
applications, 2002.
https://arxiv.org/pdf/math/0211159
Weighted Sobolev. There is a universal S_1 such that, whenever
0 <= t < min{1, T_max},
||f||_{L^4}^2 <= S_1 int_M (4|nabla f|^2 + (R + 1) f^2) dmu (30)
for every Lipschitz f on (M, g(t)).
Proof. Initial mu bound. For normalized f, set y = int |nabla f|^2.
The initial Sobolev inequality (19) gives
int f^2 log f^2 <= log int f^4 <= 2 log( S (y + 1) ). (31)
Substitute (31) in (28) with parameter s and use int (4|nabla f|^2 +
R f^2) >= 4 y (since R >= 0):
4 s y - 2 log(S (y + 1)) - 2 log(4 pi s) - 4
= 4 s y - 2 log(s(y+1)) - 2 log(4 pi S) - 4. (32)
Writing z = s(y + 1) >= s > 0, the function 4z - 2 log z has minimum
2 + 2 log 2 at z = 1/2. So (32) is bounded below by 2 + 2 log 2 -
2 log(4 pi S) - 4, a universal constant. Thus mu(g(0), s) >= -C for
every 0 < s <= 2.
Log-Sobolev. For 0 < tau <= 1 and 0 <= t < T_max the parameter
tau + t < 2, and (29) gives mu(g(t), tau) >= -C. Unpacking (28),
int f^2 log f^2 dmu <= tau E_0(f) - 2 log(1/tau) + C, (33)
where E_0(f) = int (4|nabla f|^2 + R f^2) and C is universal, for
||f||_{L^2} = 1. Approximation by smooth positive functions extends
(33) to Lipschitz f.
Support estimate. Take normalized f with support volume m > 0.
Jensen's inequality gives int f^2 log f^2 dmu >= -log m. Setting
tau = (1 + E_0(f))^{-1} <= 1 in (33),
log(1/m) <= 1 + 2 log(1 + E_0(f)) + C, (34)
hence 1 <= C' m^{1/2}(1 + E_0(f)) for a universal C'. Homogenizing
to a general f,
||f||_{L^2}^2 <= C' m^{1/2}(E_0(f) + ||f||_{L^2}^2). (35)
Apply Lemma 1 with a = 4 and P = R + 1: inequality (35) is (13) with
E = E_0 + ||·||_{L^2}^2, so (30) follows.
Ye, The logarithmic Sobolev and Sobolev inequalities along the Ricci
flow, Comm. Math. Stat. 3 (2015) 1-36.
https://arxiv.org/pdf/0707.2424
Zhang, A uniform Sobolev inequality under Ricci flow,
Int. Math. Res. Not. 2007 (2007), rnm056.
https://arxiv.org/pdf/0706.1594
High-curvature tail. Fix a universal C_q > 0 such that, in any
orthonormal frame moving by partial_t e_i = Ric(e_i), the Riemann
tensor satisfies
partial_t Rm = Delta Rm + Q(Rm), |Q(Rm)| <= C_q |Rm|^2. (36)
The frame motion absorbs the Ric-linear terms of the standard Riemann
evolution. Kato's inequality applied to u = |Rm| gives
(partial_t - Delta) u <= C_q u^2 where u > 0. (37)
Threshold. Define K on [0, T) by
K(0) = 12 C_0, K'(t) = C_q K(t)^2, T = min{1/12, 1/(24 C_0 C_q)}. (38)
Then 12 C_0 C_q T <= 1/2, and K(t) = 12 C_0/(1 - 12 C_0 C_q t) is
bounded by 24 C_0 on [0, T]. Set w(t) = (u(t) - K(t))_+.
Tail propagation. After discarding finitely many j, there are
uniform S_2 and C such that, for all 0 <= t < min{T, T_max},
||f||_{L^4}^2 <= S_2 int (|nabla f|^2 + f^2) dmu, (39a)
int w(t)^2 dmu <= e^{C t} int w(0)^2 dmu, (39b)
int |Rm(t)|^2 dmu <= C. (39c)
Moreover sup_t ||w(t)||_{L^2} tends to zero.
Proof. Bootstrap. Suppose ||w(t)||_{L^2} <= eta for a small eta to be
chosen. Since R >= 0 and R <= C|Rm|, R <= C(K_max + w). The weighted
Sobolev (30) reads
||f||_{L^4}^2 <= S_1 int (4|nabla f|^2 + C_1 f^2) dmu
+ S_1 C int w f^2 dmu. (40)
By Hölder, int w f^2 <= eta ||f||_{L^4}^2. Choosing eta so that
2 S_1 C eta <= 1 makes the coefficient of ||f||_{L^4}^2 on the left at
least 1/2, and (39a) holds with S_2 = 2 S_1 max(4, C_1).
Tail energy. On {u > K}, (37) and K' = C_q K^2 give
(partial_t - Delta) w <= C_q (w^2 + 2 K w). (41)
Combining with partial_t dmu = -R dmu <= 0 and integrating by parts
against 2w, for Y(t) = int w(t)^2 dmu,
Y'(t) <= -2 int |nabla w|^2 + 2 C_q int w^3
+ 4 C_q K_max int w^2. (42)
By Hölder and (39a), int w^3 <= ||w||_{L^4}^2 ||w||_{L^2} <=
eta S_2 int (|nabla w|^2 + w^2). Choosing eta so 2 C_q eta S_2 <= 1,
Y'(t) <= -int |nabla w|^2 + C Y(t) <= C Y(t). (43)
Initial smallness. From (2) and (3b),
int w(0)^2 dmu <= C_0^2 int (R - 12)_+^2 dmu
<= 16 C_0^2 ||E||_{L^2}^2 -> 0. (44)
Bootstrap closure. For large j, Y(0) e^{C T} < eta^2. Gronwall on
any sub-interval where the bootstrap holds gives Y(t) <= e^{C t} Y(0)
< eta^2, closing the bootstrap by continuity and proving (39b).
Energy bound. Since |Rm| <= K + w and (a+b)^2 <= 2a^2 + 2b^2,
int |Rm|^2 dmu <= 2 K_max^2 V_0 + 2 e^{C T} Y(0) <= C, (45)
giving (39c).
Chan, Chen and Lee, Small curvature concentration and Ricci flow
smoothing, J. Funct. Anal. 282 (2022) 109420.
https://doi.org/10.1016/j.jfa.2022.109420
Noncollapse. There is a universal v_* > 0 with
Vol B(x, r) >= v_* r^4 for every x in M and 0 < r <= 1. (46)
Proof. Insert into (39a) the cutoff that is 1 on B(x, r/2), 0 off
B(x, r), and has |nabla phi| <= 2/r:
V(r/2)^{1/2} <= S_2 (4/r^2 + 1) V(r) <= 5 S_2 V(r) / r^2. (47)
Writing a(r) = V(r)/r^4, this is a(r) >= c a(r/2)^{1/2} with
c = 1/(20 S_2). Iterating k times,
a(r) >= c^{2 - 2^{1-k}} a(2^{-k} r)^{2^{-k}}. (48)
Smoothness at fixed time gives a(s) -> omega_4 as s -> 0+, so the
right side tends to c^2 as k -> infinity. Thus a(r) >= c^2, and
v_* = c^2 > 0.
Lifespan. For all large j, the flow g_j(t) exists on [0, T]. With
A = 8 K_max T + 4 (universal, positive), for every existing
0 < t < T,
max_M |Rm(t)| <= A / t. (49)
Proof. Choose A > 2 with A/T >= 1 and A/T >= 4 K_max. Suppose that
s max_M |Rm(s)| first reaches A at some existing s = t < T, so
Q = max_M |Rm(t)| = A/t. Then Q >= A/T >= 4 K(t). The first-crossing
hypothesis gives max_M |Rm(s)| < A/s for s in [t - Q^{-1}, t]. Since
Q = A/t and A >= 2, t - Q^{-1} >= t/2, so max_M |Rm(s)| < 2 Q
throughout that slice.
Shi's estimate on the parabolic slab, applied after the rescaling
g_tilde(s) = Q g(t - s/Q), gives |nabla Rm|(t) <= C Q^{3/2} at every
point. Let p realize the curvature maximum Q at time t. On
B(p, c Q^{-1/2}) with c universal, |Rm| >= (3/4) Q, so w(t) >= Q/2
there. Since Q >= 4 K(t) and Q >= 1, and c Q^{-1/2} <= 1, the
noncollapse bound gives
Vol B(p, c Q^{-1/2}) >= v_* c^4 Q^{-2}, (50)
so int w(t)^2 dmu >= v_* c^4/4. This contradicts the tail decay
sup ||w(t)||_{L^2} -> 0 for large j. So no crossing occurs and (49)
holds on every existing [0, min{T, T_max}).
If T_max <= T, then (49) controls |Rm| on [T_max/2, T_max), and a
compact-interval curvature bound controls it on [0, T_max/2], so
|Rm| is bounded on [0, T_max), contradicting the curvature
continuation criterion. Hence T_max > T for large j.
Einstein defect. Let lambda(t) = 3/(1 - 6t) on [0, T] and set
L = Ric - lambda(t) g. Since T <= 1/12, lambda is smooth on [0, T],
bounded by 6, with lambda(0) = 3 and lambda'(t) = 2 lambda(t)^2. At
t = 0, the initial energy ||L||_{L^2} = ||Ric - 3g||_{L^2} tends to
zero by (3e).
Proposition 2. After discarding finitely many j,
sup_{0 <= t < T} ||L(t)||_{L^2} -> 0. (51)
Proof. In the moving frame partial_t e_i = Ric(e_i), the metric
components are constant, so Delta g = 0 and Rm(g) = Ric. The Ricci
tensor satisfies
partial_t Ric = Delta Ric + 2 Rm(Ric), (52)
where (Rm(B))_{ij} = sum_{k,l} R_{ikjl} B_{kl} on symmetric 2-tensors B.
The frame motion absorbs the -2 Ric^2 term of the fixed-frame
evolution. Combined with lambda' = 2 lambda^2,
partial_t L = Delta L + 2 Rm(L) + 2 lambda L. (53)
Reaction bounds. On [0, T], lambda <= 6 and |Rm| <= K_max + w, so
|2 lambda L + 2 Rm(L)| <= C (1 + w) |L|. (54)
Energy. Using partial_t dmu = -R dmu <= 0,
d/dt ||L||^2 <= -2 int |nabla L|^2 + 2 C int (1 + w) |L|^2. (55)
Hölder and (39a) applied to |L| (with Kato's inequality
|nabla |L|| <= |nabla L|) give
int w |L|^2 <= eta_j S_2 int (|nabla L|^2 + |L|^2), (56)
where eta_j = sup_{[0, T)} ||w||_{L^2} -> 0. For 2 C eta_j S_2 <= 1,
d/dt ||L||^2 <= C ||L||^2, (57)
which by Gronwall gives ||L(t)||^2 <= e^{C T} ||L(0)||^2 -> 0.
Sectional defect. On the finite-dimensional space of algebraic
curvature tensors over R^4, define
F(B) = max{0, sup_{orthonormal X, Y} (-B(X,Y,X,Y))}. (58)
Then F(Rm) = 0 exactly when sec >= 0, and otherwise is the magnitude
of the most negative sectional curvature. F is nonnegative, convex,
orthogonally invariant, positively homogeneous of degree 1, and
1-Lipschitz. Its convolution F_delta with a smooth nonnegative radial
mollifier at scale delta is smooth, nonnegative, convex, orthogonally
invariant, and satisfies
|DF_delta| <= 1, |F_delta - F| <= C delta. (59)
Proposition 3. There is a universal C with
int_M F(Rm(t)) dmu <= C t, 0 <= t < T. (60)
Proof. In the moving frame,
(partial_t - Delta) F_delta(Rm) = DF_delta(Rm)[(partial_t - Delta) Rm]
- sum_a D^2 F_delta(Rm)[nabla_a Rm, nabla_a Rm]. (61)
The Hessian term is nonnegative by convexity, and
(partial_t - Delta) Rm = Q(Rm) by (36), so
(partial_t - Delta) F_delta(Rm) <= |DF_delta| |Q(Rm)| <= C_q |Rm|^2. (62)
Energy. With partial_t dmu = -R dmu <= 0 and F_delta >= 0,
d/dt int F_delta(Rm) dmu <= C_q int |Rm|^2 dmu <= C_q C. (63)
Since sec(g) >= 0 at time 0, F(Rm(0)) = 0, and
int F_delta(Rm(0)) dmu <= C delta V_0. Integrating in time,
int F_delta(Rm(t)) dmu <= C delta V_0 + C_q C t. (64)
Letting delta -> 0 by dominated convergence gives (60).
Compact limit. Fix t_* = T/2 in (0, T). From (49), |Rm| <= A/t_* on
M_j at t_*. From (46), Vol B(x, r) >= v_* r^4 for r <= 1. A maximal
set of 1/2-separated points along a geodesic of length diam has at
most 16 V_0/v_* elements, since each 1/2-ball has volume at least
v_*/16, so
diam(M_j, g_j(t_*)) <= 32 V_0/v_* =: D_*. (65)
The Cheeger-Gromov-Taylor injectivity radius estimate applied with
|Rm| <= A/t_* and Vol B(x, 1) >= v_* gives
inj(M_j, g_j(t_*)) >= i_0 > 0, (66)
with i_0 uniform.
Cheeger, Gromov, Taylor, Finite propagation speed, kernel estimates
for functions of the Laplace operator, and the geometry of complete
Riemannian manifolds, J. Differential Geom. 17 (1982) 15-53.
https://doi.org/10.4310/jdg/1214436699
Hamilton compactness. With global curvature bound |Rm| <= A/t on
(0, T) and injectivity radius i_0 at t_*, Hamilton's compactness
theorem for Ricci flow, together with the local version in the book
of Morgan and Tian, gives after passing to a subsequence a complete
smooth Ricci flow (N, h(t)) defined for t in (0, T), together with
time-independent embeddings
Phi_j: (N, h(t_*)) -> (M_j, g_j(t_*)), (67)
such that (Phi_j)^* g_j(t) -> h(t) smoothly on compact subsets of
N x (0, T).
Hamilton, A compactness property for solutions of the Ricci flow,
Amer. J. Math. 117 (1995) 545-572.
https://doi.org/10.2307/2375080
Morgan and Tian, Ricci Flow and the Poincare Conjecture, Clay Math.
Monographs 3, AMS/CMI 2007.
https://bookstore.ams.org/cmi-3
Topping, Remarks on Hamilton's compactness theorem for Ricci flow,
J. reine angew. Math. 692 (2014) 173-191, discusses the completeness
conditions under which compactness holds; the bounds above are global
in space, so no completeness issue arises.
https://doi.org/10.1515/crelle-2012-0078
Compactness of N. If N were noncompact, then for every L > 2 D_*
there would be a point q at h(t_*)-distance L from a base point p.
For large j, Phi_j is defined on B(p, L) and its pullback of h(t_*)
is within a bounded factor of g_j(t_*). Any curve in M_j from
Phi_j(p) to Phi_j(q) would have length at least L/2 > D_*, contra-
dicting (65). So N is compact.
Eventual diffeomorphism. Since N is compact, Phi_j is defined on all
of N for large j. Its image is open by the inverse function theorem
and closed by compactness of M_j; connectedness of M_j then forces
the image to be all of M_j. So Phi_j: N -> M_j is a diffeomorphism,
and N inherits simple connectivity from M_j.
Einstein limit. Smooth convergence together with (51) gives
Ric_{h(t)} = lambda(t) h(t) for 0 < t < T. Since h is itself a Ricci
flow, partial_t h = -2 Ric_h = -2 lambda h, so h(t) = c(t) h(t_*)
with c(t_*) = 1 and c' = -2 lambda c. Solving, c(t) = (1 - 6t)/
(1 - 6 t_*), and defining h = h(t_*)/(1 - 6 t_*) yields h(t) = (1 - 6t) h.
Constant rescaling of a metric preserves its Ricci tensor: Ric_{c g} = Ric_g.
Hence Ric_h = lambda(t) (1 - 6t) h = 3h.
Nonnegative sec in the limit. For every fixed t in (0, T), (60) and
smooth convergence give
int_N F(Rm_{h(t)}) dmu_{h(t)} <= C t. (68)
Under a constant rescaling g -> c g with c > 0, an orthonormal pair
X, Y for c g is c^{-1/2} times an orthonormal pair for g, and
Rm_{c g}(X, Y, X, Y) = c^{-1} Rm_g(X, Y, X, Y). So F(Rm_{c g}) =
c^{-1} F(Rm_g). Applied to h(t) = (1 - 6t) h, together with
dmu_{h(t)} = (1 - 6t)^2 dmu_h,
(1 - 6t) int_N F(Rm_h) dmu_h <= C t. (69)
Letting t -> 0+, int_N F(Rm_h) dmu_h = 0. Since F(Rm_h) >= 0 is
continuous, F(Rm_h) = 0, and sec_h >= 0 everywhere on N.
Compactness statement. Call (M_j, g_j) admissible when it satisfies
the hypotheses of the main theorem with D(g_j) -> 0. For every
admissible sequence and every sufficiently large j, there are a
closed connected simply connected four-manifold N with an Einstein
metric h satisfying Ric_h = 3h and sec_h >= 0, and diffeomorphisms
Phi_j: N -> M_j.
Gap theorem. If no universal epsilon_0 existed, then for every m
there would be an admissible (M_m, g_m) with D(g_m) < 1/m and M_m
diffeomorphic to none of S^4, CP^2, S^2 x S^2. The compactness
statement yields a subsequence diffeomorphic to a fixed (N, h) with
Ric_h = 3h and sec_h >= 0. The exact-Einstein classification under
nonnegative sectional curvature identifies N as one of the three
standard manifolds, contradiction. So a universal epsilon_0 > 0
exists.
The classification premise is the exact-Einstein classification
under nonnegative sectional curvature: every closed connected
four-manifold (N, h) with Ric_h = 3h and sec_h >= 0 has universal
Riemannian cover isometric to one of the round S^4 of sectional
curvature 1, the Fubini-Study CP^2 normalized so Ric = 3g, or
S^2(1/sqrt3) x S^2(1/sqrt3). Partial results appear in Gursky-LeBrun,
Cao-Tran and Liu, cited earlier. The complete classification is the
subject of a separate recent preprint. The argument above uses the
classification only on the exact Einstein limit, never on the
original metrics.
The integral-curvature diameter estimates of Petersen and Sprouse,
and Sprouse, develop Myers-type arguments more systematically under
L^p curvature bounds; the segment inequality (9) suffices here.
Petersen and Sprouse, Integral curvature bounds, distance estimates
and applications, J. Differential Geom. 50 (1998) 269-298.
https://doi.org/10.4310/jdg/1214461171
Sprouse, Integral curvature bounds and bounded diameter,
Comm. Anal. Geom. 8 (2000) 531-543.
https://doi.org/10.4310/CAG.2000.v8.n3.a4
The proof uses no auxiliary bound on curvature, volume, diameter,
injectivity radius, Sobolev constant, or derivative of curvature. The
nonnegativity of sectional curvature is a hypothesis at time zero
only. The recovery of sec_h >= 0 in the limit does not require that
sec_{g(t)} >= 0 along the flow, nor that the original metrics converge.